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Let $V\subset\mathbb{P}^n(\overline{F}_{\hskip-0.7mm q})$ be a completeintersection defined over a finite field $F_{\hskip-0.7mm q}$ of dimension $r$and singular locus of dimension at most $0\le s\le r-2$. We obtain an explicitversion of the Hooley--Katz estimate $||V(F_{\hskip-0.7mmq})|-p_r|=\mathcal{O}(q^{(r+s+1)/2})$, where $|V(F_{\hskip-0.7mm q})|$ denotesthe number of $F_{\hskip-0.7mm q}$-rational points of $V$ and$p_r:=|\mathbb{P}^r(F_{\hskip-0.7mm q})|$. Our estimate improves all theprevious estimates in several important cases. Our approach relies on tools ofclassical algebraic geometry. A crucial ingredient is a new effective versionof the Bertini smoothness theorem, namely an explicit upper bound of the degreeof a proper Zariski closed subset of $(\mathbbP^{n})^{s+1}(\overline{F}_{\hskip-0.7mm q})$ which contains all the singularlinear sections of $V$ of codimension $s+1$.

Explicit Estimates for the Number of Rational Points of Singular Complete Intersections over a Finite Field